(1)解法一:连接OC,∵OA是⊙P的直径,∴OC⊥AB,
在Rt△AOC中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152233841044.png)
,
在 Rt△AOC和Rt△ABO中,∵∠CAO=∠OAB
∴Rt△AOC∽Rt△ABO,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223400751.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223400638.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223415639.png)
, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223431691.png)
解法二:连接OC,因为OA是⊙P的直径, ∴∠ACO=90°
在Rt△AOC中,AO=5,AC=3,∴OC=4,
过C作CE⊥OA于点E,则:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223462958.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152234627250.png)
即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223478921.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223493630.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152235091440.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223525759.png)
,
设经过A、C两点的直线解析式为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223540589.png)
.
把点A(5,0)、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223525759.png)
代入上式得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152235711123.png)
, 解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223587972.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223603780.png)
, ∴点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223634729.png)
.4分
(2)点O、P、C、D四点在同一个圆上,理由如下:
连接CP、CD、DP,∵OC⊥AB,D为OB上的中点, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223649781.png)
,
∴∠3=∠4,又∵OP=CP,∴∠1=∠2,∴∠1+∠3=∠2+∠4=90°,
∴PC ⊥CD,又∵DO⊥OP,∴Rt△PDO和Rt△PDC是同以PD为斜边的直角三角形,∴PD上的中点到点O、P、C、D四点的距离相等,
∴点O、P、C、D在以DP为直径的同一个圆上; ···········6分
由上可知,经过点O、P、C、D的圆心
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223649320.png)
是DP的中点,圆心
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223681783.png)
,
由(1)知:Rt△AOC∽Rt△ABO,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223696742.png)
,求得:AB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223712456.png)
,在Rt△ABO中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152237271184.png)
,OD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223743941.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223759736.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230152237741002.png)
,点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223649320.png)
在函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223259527.png)
的图象上,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223821908.png)
, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223837885.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823015223852169.png)
.